Nonstandard Analysis

Preliminary remarks: In the following, we shall use the definitions established in the chapter on Set Theory, and we will usually take m, n ∈ ωℕ*. We seek to study integration and differentiation on an arbitrary non-empty subset A of the sets (ω)ℂn or (ω)ℝn for arbitrary n, which we shall combine with the notation (ω)ℍn = (ω)ℍ× ... ×(ω)ℍ, where each ℍ represents either ℂ or ℝ in some arbitrary order, since we shall not consider quaternions here. In particular, we will consider inconcrete and infinite sets that are conventionally non-measurable, and functions that are conventionally discontinuous. Every element not in the image set is replaced by the neighbouring element in the target set, and if multiple choices are possible one single choice is selected. If we do not do this, the mapping is not (meaningfully) defined. This can easily be generalised to other sets.

Definition: The function ||·||: V → (ω)ℝ>0 where V is a vector space over (ω)ℍ is called a norm, if for all x, y ∈ V and λ ∈ (ω)ℍ, we have that: ||x|| = 0 ⇒ x = 0 (definiteness), ||λx|| = |λ| ||x|| (homogeneity), and ||x + y|| ≤ ||x|| + ||y|| (triangle inequality). The dimension of V is defined as the maximal number of linearly independent vectors, and is denoted by dim V. The norms ||·||a and ||·||b are said to be equivalent if there exist non-infinitesimal σ, τ ∈ cℝ>0 such that, for all x ∈ V, it holds that:

σ||x||b ≤ ||x||a ≤ τ||x||b.

Theorem: Let N be the set of all norms in V. Every norm on V is equivalent if and only if ||x||a/||x||b is finite but not infinitesimal for all ||·||a, ||·||b ∈ N and all x ∈ V*.

Remark: On cℍn with n ∈ ωℕ*, all norms are equivalent if we disallow the definition of infinite or infinitesimal norm values. In the following, ||·|| denotes the Euclidean norm. Results can easily be generalised to equivalent norms.

Definition: We describe neighbouring points in A by means of the irreflexive symmetric neighbourhood relation B ⊆ A2. The set ÑB(z0) of all neighbours of z0 ∈ A (with respect to B in A) is called the neighbourhood of z0 (with respect to B in A). The function γ: C → A ⊆ ℂn where C ⊆ ℝ is h-homogeneous and h is infinitesimal, is called a path, if ||γ(x) – γ(y)|| is infinitesimal for all neighbouring points x, y ∈ C and (γ(x), γ(y)) ∈ B. The neighbourhood relations of B are systematically written as (predecessor, successor) with the notation (z0, ↷z0) or (↶z0, z0), where ↷ is pronounced "succ" and ↶ is pronounced "pre". This applies analogously to the neighbourhood relation D ⊆ C2.

Definition: Let z0 ∈ A ⊆ ℍn and f: A → (c)ℍm. In the following, we will omit proofs for predecessors, since they are analogous to the proofs for successors. We say that f is αB-successor-continuous in z0 in the direction ↷B z0 if for infinitesimal α ∈ (ω)ℝ>0, we have that

||f(↷B z0) - f(z0)|| < α.

In general, when ||x – y|| < α for x, y ∈ A, we write x ≈α y, meaning "α-infinitesimally equal". If the exact modulus of α does not matter, α can be omitted in the notation. If f is αB-successor-continuous for all z0 and ↷B z0 ∈ ÑB(z0), we simply say that it is αB-continuous. We say that α is the degree of continuity. If the inequality only holds for α = 1/c, we simply say that f is (B-successor-)continuous. The property of αB-predecessor-continuity is defined analogously.

Remark: In practice, α is chosen by estimating f (for example after considering any jump discontinuities). If B is obvious or irrelevant, it can be omitted. Below, we systematically omit it when B = (ω)ℍ2n.

Example: The function f: ℝ → {±1} with f(x) = (-1)x/d0 is nowhere successor-continuous on ℝ, but its absolute value is (cf. Transcendental Numbers). Here, x/d0 is an integer since ℝ is d0-homogeneous. If we set f(x) = 1 for rational x and = -1 for irrational x, then f(x) is partially d0-successor-continuous on irrational numbers, unlike the conventional notion of continuity.

Definition: For f: A → (ω)ℍm, we define

d↷B zf(z) := f(↷B z) - f(z)

to be the B-successor-differential of f in the direction ↷B z for z ∈ A. If dim A = n, then we can specify d↷B zf(z) d↷B zf(z) as d((↷B)z1, ... , (↷B)zn)f(z). If f is the identity, i.e. f(z) = z, then we can write d↷B zBz instead of d↷B zf(z). If A or ↷B z is obvious or irrelevant, it can be omitted. The conventionally real case can be defined analogously to the above.

Remark: If the modulus of the B-successor-differential of f in the direction ↷B z at z ∈ A is smaller than α and infinitesimal, then f is also αB-successor-continuous at that point.

Definition: The m arithmetic means of all fk(↷B z) of f(z) give the m averaged normed tangential normal vectors of m (uniquely determined) hyperplanes, defining the mn continuous partial derivatives of the Jacobian matrix of f, which is not necessarily continuous. The hyperplanes are taken to pass through fk(↷B z) and f(z) translated towards 0. We can minimise the moduli of their coefficients by solving a very simple linear programme (cf. Linear Programming).

From this we obtain together with the definition of convexity of functions directly the

Theorem: Every in A ⊆ (ω)ℍn convex resp. concave function f: A → (ω)ℝ is αB-successor-continuous and B-successor-differentiable.⃞

Theorem: An arbitrary mapping f: X → X on an arbitrary set X is bijective if it is either injective or surjective.

Proof: The claim follows directly from the fact that all (pre-)images are pairwise distinct.⃞

Remark: Note that this theorem does not apply to the successor function s in ωℕ, since s: ωℕ → ωℕ* ∪ {|ωℕ|}.

It is clear that g is fully specified by this definition, and continuous. Now let the function Φ: I = [0, 1] → ℝ2 be defined by

"

The function Φ is at least continuous since the sums are ultimately locally linear functions in t, when ∞ is replaced by ώ. It would however be an error to believe that [0, 1] can be bijectively mapped onto [0, 1]2, in this way, e.g. because the powers of four in g, and the values 0 and 1 taken by g in two sub-intervals thin out [0, 1]2 so much that a bijection is clearly impossible. Restricting the proof to rational points only is simply insufficient.

Definition: A point x (↷x) of a function f: A ⊆ ωℝ → ωℝ is said to be a right (left) jump discontinuity of s:= |f(↷x) – f(x)| upwards (downwards), or vice versa if s > 1/|ωℕ*|.

Proof: Between -|ωℕ*| and |ωℕ*|, at most |ωℕ*||ωℤ*| jump discontinuities with a jump of 1/|ωℕ*| are possible, totalling a maximum of |ωℤ*| - 1 when all viewed together. If the function does not decrease at non-discontinuities, like a step function, then the claim follows after correctly accounting for the ends of the number line ωℝ.⃞

Remark: This theorem corrects Froda's theorem and makes it more precise. If we precede each set by the superscript c, we obtain the statement for conventional sets.

Definition: The partial derivative in the direction ↷B zk of F: A → (ω)ℍ at z = (z1, ..., zn) ∈ A ⊆ (ω)ℍn with k ∈ [1, n]ℕ is defined as

With this notation, if the function f satisfies f = (f1, ..., fn): A → (ω)ℍn with z ∈ A ⊆ (ω)ℍn

then f(z) is said to be exact B-successor-derivative F′↷B z B(z) or the exact B-successor-gradient grad↷B z of the function F at z, which is said to be exactly B-differentiable at z in the direction ↷B, provided that each quotient exists in (ω)ℍ. ∇ is the Nabla operator. If this definition is satisfied for every z ∈ A, then F is said to be an exactly B-differentiable B-antiderivative of f. On the (conventionally) (infinitely) reals, the left and right B-antiderivatives Fl(x) and Fr(x) at x ∈ (ω)ℝ distinguish between the corresponding B-derivatives.

If A or ↷B z are obvious from context or irrelevant, they can be omitted. The conventional case may be obtained analogously to the above. In the case n = 1, we say that F′rB(w) is the right exact B-derivative for ↷B w > w ∈ (ω)ℝ and that F′lB(w) is the left exact derivative for ↷B w < w. If both directions have the same value, we say that F′B(z) is the exact derivative (when A = cℂ and n = 1, this reduces to the conventional case where F is holomorphic).

Remark: Clearly, the B-antiderivatives of a given function only differ by a (conventional) (infinite) complex or (infinite) real additive constant. The B-antiderivatives of discontinuous functions can typically only be derived by adding and appropriately recombining easier αB-continuous functions (e.g. by reversing the rules of differentiation).

Remark: The chain rule and the rules stated below can be extended to (infinite) complex sets and left exact derivatives. We will subsequently omit the sets and neighbourhood relations. Let f and g be right exactly differentiable (infinite) real functions at x ∈ A ⊆ (ω)ℝ.

Remark: For the product and quotient rule to be as precise as the conventional one, the arguments and function values must belong to a smaller level of infinity than 1/d0, and f and g must be sufficiently (α-)continuous at x ∈ A (i.e. α must be sufficiently small) to allow ↷x to be replaced by x. An analogous principle holds for infinitesimal arguments.

Remark: The right exact derivative of the inverse function

f-1′r(y) = 1/f′r(x)

can be derived from y = f(x) and the identity x = f-1(f(x)) using the chain rule with an equal level of precision. L'Hôpital's rule also makes sense for (α-)continuous functions f and g such that f(w) = g(w) = 0 with w ∈ A and g(↷ w) ≠ 0, and may be stated as:

Remark: Differentiability is thus easy to establish. In the (conventionally) (infinite) real case, we can set

wherever this quotient is defined. This is especially useful when ↷B w - w = w - ↶B w, and the combined derivatives both have the same sign. This definition has the advantage of allowing us to view F′bB(w) as the "tangent slope" at the point w, especially when F is αB-continuous in w. It also results in simpler rules of differentiation, in particular since a derivative value of 0 is most suitable for cases with opposite signs (see below). In other cases, we simply calculate the arithmetic mean of both exact derivatives. This can be extended to the conventional complex numbers analogously.

Definition: Given z ∈ A ⊆ (ω)ℍn, we define

to be the exact B-integral of the vector field f = (f1, ..., fn): A → (ω)ℍn on A and we say that f(z) is B-integrable. If this requires us to remove at least one point from A, then we say that the exact B-integral is improper. For γ: [a, b[C → A ⊆ (ω)ℍn, C ⊆ ℝ and f = (f1, ..., fn): A → (ω)ℍn, we say that

Definition: The function µh: A → ℝ≥0 where A ⊆ (ω)ℂn is an m-dimensional set with h ∈ ℝ>0 less than or equal the minimal distance of the points in A, m ∈ ℕ≤2n, µh(A) := |A| hm, and µh(∅) = |∅| = 0 is called the exact h-measure of A and A is said to be h-measurable, with dim (ω)ℂ = 2.

Remark: µh(A) is clearly additive and uniquely determined, i.e. if A is the union of pairwise disjoint h-homogeneous sets Ak for k ∈ ℕ, then

It is also strictly monotone, i.e. if h-homogeneous sets A1, A2 ⊆ (ω)ℍn satisfy A1 ⊂ A2 then µh(A1) < µh(A2). If h is not equal on all considered sets Ak, we choose the minimum of all h and homogenise as described in Set Theory. This measure is more precise than other measures and is optimal, since its value is neither smaller nor greater than the distances of the points parallel to the coordinate axes, as it simply considers the neighbourhoods of a point. Concepts such as σ-algebras and null sets are not required, since the only null set in this context is the empty set.

Examples: Consider the set A ⊂ [0, 1[ of points, whose least significant bit is 1 (0) in their (conventionally) real binary representation. Then µd0(A) = ½. Real numbers represent a further refinement of the conventionally real numbers, by dividing the conventionally real intervals into (significantly) finer sub-intervals. Since A is an infinite (conventionally uncountable) union of individual points (without the neighbouring points of [0, 1[ in A), and these points are Lebesgue null sets, A is not Lebesgue measurable, however it is exactly measurable. Similarly, consider the subset S of [0, 1[ × [0, 1[ of all points with least significant bit 1 (0) in both coordinates. This set has exact measure µd0(S) = ¼.

Remark: Binary representation is also suitable for BBP series and Euler summation - or at least some power of two (8 is ideal, cf. Calculation of Times).

Standard estimate: For n = 1, and M = max |f(z)| on γ, by successively applying the triangle inequality, we have that

where the right-hand sum is known as the Euclidean path length L(γ), including in the case n > 1.

Definition: The exact standard measure is µd0.

Remark: In the standard estimate, the conventionally real case is defined analogously to the above. If it is clear that we deal with the standard measure, we may omit d0. The value of the exact line integral on (c)ℍ is usually consistent with the conventional path integral; however, f does not need to be continuous and the proper B-line integral exists always. It can easily be seen that the exact B-line integral is linear and monotone in the (conventional) (infinite) real case. The art of integration lies in correctly combining the summands of the defining sum.

Remark: Because ℂ and ℝ2 are isomorphic, we have something similar in the complex case and

Remark: An alternative definition of the exact volume integral may be given as described further below. However, the original definitions are easier to manipulate. In some cases, suitable Landau notation may be useful. If the result of differentiation lies outside of the domain, it should be replaced by the closest number within the domain. If this is not uniquely determined, the result can either be given as the set of all such numbers, or we can select the preferred result (e.g. according to a uniform rule).

Example: Let [a, b[hωℤ be a non-empty h-homogeneous subset of [a, b[ωℝ, and write B = ]a, b]hωℤ. For h = 1/c and c = -a = b – h, [a, b[hωℤ is comparable with cℝ. Now let Tr be a right B-antiderivative of a not necessarily convergent Taylor series t on [a, b[hωℤ and define f(x) := t(x) + ε(-1)x/h for conventionally real x and ε ≥ 1/c. For h = 1/c, f is nowhere continuous, and thus is conventionally nowhere differentiable or integrable on [a, b[hωℤ, but for all h we have the following exact result

Remark: The sets Q and C are conventionally not measurable. Thus, exact integration is more general than Riemann, Lebesgue(-Stieltjes) integration and other types of integral, which only exist on conventionally measurable sets, and function values may skip values, which are not measured then. We only consider simple examples here for purposes of illustration, and more complex examples can of course be found.

Definition: A sequence (ak) with members ak is a mapping from (ω)ℤ to (ω)ℂm: k ↦ ak. A series is a sequence (sk) with m ∈ (ω)ℤ and partial sums

Remark: Thus, in particular, the Riemann series theorem is false, since when summing positive summands to a desired value we are forced to add negative values until the original sum is attained, and vice versa. The same is true in the case that we obtain a smaller or larger value than the sum of the positive or negative terms, since the remainders almost fully cancel, and so on. We must avoid choosing arbitrary methods to deal with infinities if we wish to avoid making mistakes. We cannot make the mistake of thinking that something no longer exists simply because it is stored at infinity.

Remark: The associative, commutative, and distributive laws imply that sums can be arbitrarily rearranged. If we take care to calculate it correctly (using Landau symbols), the B-volume integral satisfies the following theorem:

Fubini's theorem: For A1, A2 ⊆ (ω)ℍ, f: A1×A2 → (ω)ℍ satisfies

Proof: The claim follows immediately by reordering the sums corresponding to these integrals.⃞

Example: Since

by the principle of latest substitution (see below), we obtain the (improper) integral

and not

Finiteness criterion for series: The Euclidean norm of the partial sum with the largest index max k of a series (sk), where k and j are natural numbers, is finite if and only if it may be represented in the form

where the dj ∈ cℝ≥0 form a monotonically decreasing sequence.

Proof: The sum is bounded below by 0 and bounded above by d0. The claim follows directly from the ability to arbitrarily rearrange summands, sort them according to their signs and sizes, and recombine them or split them into separate sums.⃞

Example: From the alternating harmonic series, it follows that

Definition: The following rearrangement for aj, bk ∈ (ω)ℍ gives the series product and corrects the Cauchy product:

Example: For following series product (cf. [763], p. 61 f.), we have that:

Theorem: The number s := 1 - 1/c is an upper bound for arguments that yield conventionally real sums when substituted as an argument into the geometric series.

Proof: The claim follows directly from c (1 - 1/c)ώ+1 ≈ 0.⃞

Remark: With the notation of the preceding theorem, the standard proofs given in the literature show that the sum of a power series with members anxn is conventionally real whenever its radius of convergence is ≤ s/lim sup |an|1/n.

Finiteness criterion for products (analogue of the fundamental theorem): The product

Remark: Products with ak ∈ cℂ are finite, if and only if their moduli are finite. Factors with modulus < 1 must be considered in the computation by pairing off with factors > 1, e.g. by considering the product of their reciprocals.

Definition: A sequence (ak) with k ∈ (ω)ℕ*, ak ∈ (ω)ℂ and α ∈ ]0, 1/c] is said to be α-convergent to a ∈ (ω)ℂ if there exists q ∈ ℕ* satisfying |ak - a| < α for all ak with k ≥ q such that the difference max k - q is not too large. The set α-A of all such a is called set of α-limit values of (ak). An appropriately and uniquely determined representative of this set (e.g. the final value or mean value) is called the α-limit value α-a. In the special case a = 0, we say that the sequence is a zero sequence. If the inequality only holds for α = 1/c, the α- is omitted from the notation.

Remark: We will usually choose k to be maximal and α to be minimal. Conventional limit values are often only precise to less than O(1/ώ) and in general are too imprecise, since they are often e.g. (arbitrarily) algebraic (of a certain degree) or transcendental. The conventional formulation of the definition of conventional convergence, which always requires infinitely many or almost all members of the sequence to have an arbitrarily small distance from the limit value and only allows finitely many to have larger distance, needs to be extended, since otherwise only the largest index of each sequence is taken into account and considered to be relevant (cf. [813], p. 144). Only then is the monotone convergence valid (cf. [813], p. 155).

Remark: Conventional differentiation and integration loses the ability to distinguish between transcendental and algebraic in the conventional process of taking limits. This is problematic, for example when we wish to exactly determine the roots of a polynomial. Therefore, we adopt here a different (more precise) approach.

Remark: The statement that each positive number may be represented by a fully determined unique, infinite decimal fraction is baseless, since the proof of the irrationality of 2 can also be applied to infinite decimal fractions (see p. 27 f.) Furthermore, any proof claiming that, for ε ∈ (ω)ℝ>0 - in particular whenever the phrase "for all conventionally reals ε > 0" is used - there exists a real number ε/r with real r ∈ (ω)ℝ>1, is false, because we can simply set ε := ↷0, or become stuck in an infinite regression. Therefore, in the εδ-definition of the limit value (it is questionable that δ exists, p. 235 f.) and in the εδ-definition of continuity (see p. 215) (consider for example the real function that doubles every real value but is not even uniformly continuous), ε must be restricted specific multiples of ↷0.

Remark: We need not to consider uniform continuity, since in general we can set δ := ↷0 and ε accordingly larger. If the conditions are not satisfied for two function values, then the function is not continuous at that point. Thus, continuity is equivalent to uniform continuity, by choosing the largest ε from all admissible infinitesimal values. It is also easy to show that continuity is equivalent to Hölder continuity, provided that we allow infinite real constants. The same is true for uniform convergence, since we can simply choose the maximum of the indices satisfying each argument as the index that satisfies everything, and ώ is sufficient in every case. If this is not true for a given argument, then pointwise convergence also fails. Thus, uniform convergence is equivalent to pointwise convergence, by choosing the largest of all admissible infinitesimal values.

Remark: Since between any two rational numbers there are infinitely many algebraic numbers of higher degree (see Set Theory), the principle of nested intervals is invalid (cf. p. 158). The definition of the real numbers via Dedekind cuts is therefore just as unsuitable as its definition via equivalence classes of rational Cauchy sequences (see p. 29 ff.). The best definition is therefore based on homogeneity, defining the real numbers as infinite integer multiples (i.e. including integers that do not lie in ℤ) of ↷0. The above remarks illustrate why conventional analysis cannot be preserved in its existing form.

These five examples illustrate the superiority of nonstandard analysis and the strength of using infinitesimal and infinite values.

Commutativity theorem for integrals of α-limit values: Let A ⊆ (ω)ℝ be h-homogeneous and let (fj) be a sequence of integrable functions with (infinite) natural j and such that fj: A → (ω)ℝ are α1-convergent to the integrable function f: A → (ω)ℝ. Then, we have that

gives the estimate

Proof:

⃞

Remark: As long as we are carefully to calculate correctly using Landau notation, we can exchange the order of differentiation or integration and summation (even) in (divergent) series. However, the conventional approach can lead to non-negligible error propagation in subsequent calculations, e.g. if α1μd0(A) ≥ 1/ω. Whenever exchanging the order of operations, the principle of latest variable substitution must be followed, otherwise there may be discrepancies in the results. This immediately implies:

Strong permutation theorem: Arbitrary permutations of the order of (admissible) substitution of the same variables in sequences, derivatives, and integrals, yield the same results.

First fundamental theorem of exact differential and integral calculus for line integrals: The function

Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, we have with γ: [a, b[C → (ω)ℍ that

Proof:

Corollary: For a closed path γ, we have with the conditions above that

whenever f has an antiderivative F on γ.

Remark: The conventionally real case of both fundamental theorems may be established analogously. Given v, w ∈ [a, b[C, v ≠ w and γ(v) = γ(w), it may be the case that ↷B γ(v) ≠ ↷B γ(w). It should be noted that neither the integral nor the derivative is assumed to be continuous. Actual integration (as the inverse operation to differentiation) only makes sense for continuous functions if we wish to go beyond simple summation. However, if we can express the function values in the form of finitely many continuous functions whose antiderivatives may be computed in finite time, integrals may be calculated even for discontinuous functions, if necessary by appropriately applying the Euler-Maclaurin summation formula and other simplification techniques.

Remark: The higher the number of elements included in the integration, the greater the deviation in the value of the integral can become, even with identical interval limits. When we use alternative exact differentiation, the formulas change accordingly: the more continuous the functions in question, the smaller the deviation in the value. Here, and in general, choosing a suitable rounding strategy can be helpful.

Definition: The tightened (right) exact B-integral according to the trapezoid rule is defined by

The tightened (right) exact B-integral according to the midpoint rule assuming that (z + ↷B z)/2 exists - is defined by

Remark: Since this tightened exact B-integral is clearly independent of the direction, it may be (implicitly) used to justify theorems that cancel integral values in opposite directions, such as Green's theorem (see below). In the first fundamental theorem, the derivative dB(F(z))/dBz can be tightened to the arithmetic mean (f(z) + f(↷B z))/2 resp. (f(z + ↷B z)/2), and similarly, in the second fundamental theorem, F(γ(b)) - F(γ(a)) can be tightened to (F(γ(b)) + F(↶B γ(b)))/2 - (F(γ(a)) + F(↷B γ(a)))/2 resp. F((γ(b) + ↶B γ(b))/2) - F((γ(a) + ↷B γ(a))/2), which yields approximately the original results when f and F are sufficiently α-continuous at the boundary. Since the exact B-integral is defined according to a rectangular rule, we can estimate the corresponding overall error relative to exact integration (see the literature on Numerical Mathematics).

Remark: We are integrating in the complex plane over a path whose start and end points are the limits of integration. If ↷D a(x) ≠ a(↷B x), then the final summand must be multiplied by (↷D a(x) - a(x))/(a(↷B x) - a(x)), and if ↷D b(x) ≠ b(↷B x), then the penultimate summand must be multiplied by (↷D b(x) - b(x))/(b(↷B x) - b(x)).

for g(ζ) = (f(ζ) - f(z))/(ζ - z), and so in particular when g has an antiderivative on γ([a, b[).

Proof: The claim follows directly from the corollary to the second fundamental theorem.⃞

Remark: The winding number is 0 if the path does not wind around z (antiderivative ln(ζ - z)). For n ∈ ℕ, the winding number is n (-n) if the path winds n around z n times in the positive (negative) direction. This can easily be seen in the following figure, parametrising the circle λ by ζ = z + r eit, with length 2πr for t ∈ [0, 2π[ and r ∈ ℝ>0:

Remark: This example shows that the existence of the antiderivative of the integrand of a path integral and the value of this integral in general only depends on the integrand and the (oriented) path, and not on the composition of the underlying set, the interior or exterior of a path, or holomorphicity or (null-)homology as in conventional complex analysis.

Mean value equation: Given γ([0, 2π[) = ∂Br(d) with d ∈ (ω)ℂ and r ∈ (ω)ℝ>0, then if f: Br(d) → (ω)ℂ satisfies the conditions of the integral formula, we have that

Residue theorem: Given γ([a, b[) ⊆ A ⊆ (ω)ℂ, if f: A → (ω)ℂ may be represented as

with n ∈ ℕ, aij, di ∈ (ω)ℂ and di pairwise distinct, then we have

for the closed path γ: [a, b[ → (ω)ℂ.

Proof: For all j ∈ ωℕ≤n and all j ∈ ℤ \ {-1}, we have

and

⃞

Intermediate value theorem: Let f: [a, b] → (ω)ℝ be α-continuous on [a, b]. Then f(x) takes every value between min f(x) and max f(x) to a precision of < α as x ranges over [a, b]. If f is continuous on ωℝ, then it takes every value of cℝ between min f(x) and max f(x).

Proof: Between min f(x) and max f(x), there is a gapless chain of overlapping α-neighbourhoods centred around each f(x), by α-continuity of f. The second part of the claim follows from the fact that a deviation f(↷x) - f(x)| < 1/c or |f(x) - f(↶x)| < 1/c in cℝ is smaller than the conventionally maximal admissible resolution.⃞

Definition: The derivative of a function f: A → (ω)ℝ, where A ⊆ (ω)ℝ is defined to be zero if and only if 0 lies in the interval defined by the boundaries of the left and right exact derivatives.

Example: The (2d0)-continuous function f: (ω)ℝ → {0, d0} defined by

consists of only the local minima 0 and the local maxima d0, and only has the (left and right) exact derivatives ±1.

Definition: Let f: A → (ω)ℍ for A ⊆ (ω)ℍ. We say that

is the second derivative of f at z ∈ A in the direction ↷B z.

Higher (partial) derivatives are defined analogously. The number j ∈ ℕ of partial derivatives is written as an exponent after ∂, and the variables with respect to which we are differentiating are listed in the denominator, each preceded by ∂. Multiples of the same variable are indicated with an exponent corresponding to the number of times that the variable occurs. Taylor series only make sense for ώ-times α-continuously differentiable functions, due to approximating and convergence-related behaviour.

Exchange theorem: The result of multiple partial derivatives of a function f: A → (ω)ℍ is independent of the order of differentiation, provided that variables are only evaluated and limits are only computed at the end, if applicable (principle of latest substitution).

Proof: The derivative is uniquely determined: This is clear up to the second derivative, and the result follows by (transfinite) induction for higher-order derivatives.⃞

is equal to y on the left for x = 0 and 0 on the right for y = 0. Partially differentiating the left-hand side with respect to y gives 1 ≠ 0, which is the partial derivative of the right-hand side with respect to x.

is infinitesimal if and only if theCauchy-Riemann partial differential equations

are satisfied by B in both the ↷ direction and the ↶ direction.

Proof: Since we have

and dBz = dBx + i dBy for every derivative defined on A, and since

as well as the analogous formulas for v and in the ↶ direction, we have that

The assumptions allow us to neglect the final summand, and so the claim follows.⃞

Remark: The final summand may in particular be neglected whenever f is continuous. The equations for F′B(z) imply the following necessary and sufficient condition for F to be holomorphic:

Definition: A (infinitely) real-valued function with arguments ∈ (ω)ℍn is said to be convex (concave) if every value taken by the function on arguments lying between two different arguments is non-strictly below (non-strictly above) the line connecting the two values taken by the function at this arguments. The function is said to be strictly convex (concave) if we can replace "non-strictly" by "strictly" in the above.

Definition: When integrating identical paths in opposite positive and negative directions, we adopt the counter-directional rule for integrals, stating that when following the path in the negative direction, we must choose the function value of the successor of the argument if the function is too discontinuous, implying that the integral sums to 0 over both directions.

Remark: We require this convention in order to ensure that integrals that we expect to sum to zero do in fact do so. Without it, they could potentially have a (significantly) different value.

Proof: If we consider two arbitrary squares with common edge of length d0 included in one plane, then only the edges of V×V↷ are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in ∂n-1V.⃞

Remark: It is not difficult to transfer both definition and theorem to the complex numbers.

Proof: We will prove the case A := {(x, y) : r ≤ x ≤ s, f(x) ≤ y ≤ g(x)}, r, s ∈ (ω)ℝ, f, g : ∂A → (ω)ℝ wlog, since the proof is analogous for each case rotated by 90°, and every simply connected h-set is a union of such sets. We will simply show that

since the other relation may be shown analogously. Since the regions of γ where dBx = 0 do not contribute to the integral, for negligibly small t := h(u(d, g(d)) – u(c, g(c))), we have that

⃞

Remark: The choice of m depends on the required number of sets of the type specified in the above proof, the union of which yields the simply connected h-set.

Goursat's integral lemma: If f is holomorphic on a triangle Δ ⊆ (ω)ℂ but does not have an antiderivative on Δ, then

Refutation of conventional proofs based on estimation by means of a complete triangulation: The direction in which ∂Δ is traversed is irrelevant. If Δ is fully triangulated, then wlog every minimal triangle Δs ⊆ Δ must either satisfy

or

where z1, z2 and z3 represent the vertices of Δs. By holomorphicity and cyclic permutations, this can only happen for f(z1) = f(z2) = f(z3). If we consider every adjacent triangle to Δ, we deduce that f must be constant, which contradicts the assumptions. This is because the term in large brackets is translation-invariant, since otherwise we can set z3 := 0 wlog, making this term 0, in which case z1 = z2(1 ± i√3)/2 and |z1| = |z2| = |z1 – z2|. However, since every horizontal and vertical line is homogeneous on (ω)ℂ, this cannot happen, otherwise the corresponding sub-triangle would be equilateral and not isosceles and right-angled. Therefore, in both cases, |Is| must be at least |f′(z2) O(d02)|, by selecting the vertices 0, |d0| and i|d0| wlog. Denoting the perimeter of a triangle by L, then we have that |I| ≤ 4m |Is| for an infinite natural number m, and also 2m = L(∂Δ)/ |O(d02)| since L(∂Δ) = 2m L(∂Δs) and L(∂Δs) = |O(d02)|. Therefore, we have that |I| ≤ |f′(z2) L(∂Δ)2/O(d02)|, causing the desired estimate |I| ≤ |O(dBζ)| to fail, for example if |f′(z2) L(∂Δ)2| is larger than |O(d02)|.⃞

Fundamental theorem of algebra: For every non-constant polynomial p ∈ (ω)ℂ, there exists some z ∈ (ω)ℂ such that p(z) = 0.

Indirect proof: By performing an affine substitution of variables, we can reduce to the case 1/p(0) ≠ O(d0). Suppose that p(z) ≠ 0 for all z ∈ (ω)ℂ. Since f(z) := 1/p(z) is holomorphic, we have that f(1/d0) = O(d0). By the mean value inequality |f(0)| ≤ |f|γ for γ = ∂Br(0) and arbitrary r ∈ (ω)ℝ>0, and hence f(0) = O(d0), which is a contradiction.⃞

Choosing sufficiently small (transcendental) constants in the generalised Liouville theorem shows that it does not hold. However, both theorems can be saved by extending them, if we identify 1/ω with 0. In this sense, the main theorem of Cauchy's theory of functions is true and can be proven according to Dixon (as in [473], p. 228 f.), especially as the limit mentioned there shall be 0 resp. 1/r tends to 0 for r tends to ω.

The function f gives a biholomorphic, bijective mapping from the circle definition of cℂ to the highly dense complex unit disc đd, where |đd| = |cℂ| ≫ |đ|. Thus, the Riemann mapping theorem also holds for ℂ. It is of course not possible to map all of cℂ in this way. The function 1/f gives a counterexample to Picard's great theorem when ωℂ is taken to be arbitrarily dense.

Definition: A point z0 ∈ M ⊆ (ω)ℂn or belonging to a sequence (ak) for ak ∈ (ω)ℂn and an (infinite) natural number k is called a (proper) α-accumulation point of M or of the sequence, if the ball Bα(z0) ⊆ (ω)ℂn with centre z0 and infinitesimal radius α contains infinitely many points from M or infinitely many pairwise distinct members of the sequence. If the claim holds for α = 1/ω, the α-accumulation point is simply called an accumulation point.

Let p(z) = π(z - dk) with k ∈ ωℕ and z ∈ ωℂ be an infinite product with pairwise distinct zeros dk ∈ B1/ώ(0) ⊂ ωℂ (disc around 0 with radius 1/ώ), chosen in such a way that |f(dk)| < 1/ώ for a function f holomorphic on a region G ⊆ ωℂ and that f(0) = 0. Suppose that G contains B1/ώ(0) completely. This can always be achieved by means of coordinate transformations provided that G is sufficiently "large".

Then the coincidence set {ζ ∈ G : f(ζ) = g(ζ)} of the function g(z) := f(z) + p(z), which is also holomorphic on G, contains an accumulation point at 0, and f ≠ g, contradicting the statement of the identity theorem. Examples of such f include functions with a zero at 0 that are restricted to B1/ώ(0) and simultaneously holomorphic on G. Since p(z) can take every conventional complex number, the deviation between f and g is non-negligible.

The identity theorem is also contradicted by the fact that all derivatives v(n)(z0) = w(n)(z0) of two functions v and w can be equal at a point z0 ∈ G for all n, but that v and w can be significantly different further away from this local behaviour without ceasing to be holomorphic, since not every holomorphic function can be (uniquely) developed into a Taylor series due to the approximation (of differentiation) and computation with Landau symbols (see Transcendental Numbers).

Extending to k ∈ ℕ in π(z - dk) allows entire functions with an infinite natural number of zeros to be constructed. The set of zeros is not necessarily discrete. Thus, the set of all functions that are holomorphic on a region G may contain zero divisors. Functions such as polynomials with n > 2 pairwise distinct zeros once again give counterexamples to Picard's little theorem, since they omit at least n - 1 values in ωℂ.

Remark: From the identity

for real or complex x, by differentiating, we deduce:

when the modulus of x, dx or 1/dx have different orders of magnitude.

For sufficiently small but not excessively small x, and sufficiently large but not excessively large (infinite) n, this formula can be further simplified to -1/(1+x)2, and also remains valid when x ≥ 1 is not excessively large. By successively multiplying s(j)(x) by x for j ∈ ωℕ* and subsequently differentiating, we can derive other formulas for s(j+1)(x), providing an example of divergent series that has previously not always been correctly calculated.

For all complex z, s(0)(z) holds. In the problematic special case of z = -1, this follows from L'Hôpital's rule. However, if we integrate s(0)(-x) from 0 to 1 and set n := ώ, we obtain an integral expression for ln ώ + γ in terms of Euler's constant γ. Substituting y := 1 - x, by the binomial series we obtain a series with almost exclusively infinite coefficients; if we also express ln ώ as a series, we even obtain an expression for γ.

If we illegitimately simplify the numerator of s(0)(z), we risk finding incorrect results, especially when |z| ≥ 1. So s(0)(-eiπ) is e.g. 0 for odd n, and 1 for even n, but not ½.

Theorem: Using the digamma function ψ, we have for n ∈ ω2ℕ* and small h ∈ ]0, 1] that